Matrices · Question 5

Let I= pmatrix1 & 0\ 0 & 1 pmatrix and P= pmatrix2 & 0\ 0 & 3 pmatrix. Let Q= pmatrixx & y\ z & 4 pmatrix for some non-zero real numbers x, y, and z, for which there is a 2 2 matrix R with all entries being non-zero real numbers, such that QR=RP. Then which of the following statements is (are) TRUE?

1. The determinant of Q-2I is zero
2. The determinant of Q-6I is 12
3. The determinant of Q-3I is 15
4. yz=2

Answer

(1) The determinant of Q-2I is zero, (2) The determinant of Q-6I is 12

Solution

Let R=[ r1 r2 ] (columns, both non-zero). QR=RP Qr1=2r1, Qr2=3r2, so 2 and 3 are the eigenvalues of Q.

Trace: x+4=5 x=1; determinant: 4x-yz=6 yz=-2.

(Q- I)=( -2)( -3).

=2: 0 (true); =6: 4 3=12 (true); =3: 0 15; yz=-2 2.

Answer: (A), (B)